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Rotational invariant estimator for general noisy matrices

Author

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  • Joel Bun
  • Romain Allez
  • Jean-Philippe Bouchaud
  • Marc Potters

Abstract

We investigate the problem of estimating a given real symmetric signal matrix $\textbf{C}$ from a noisy observation matrix $\textbf{M}$ in the limit of large dimension. We consider the case where the noisy measurement $\textbf{M}$ comes either from an arbitrary additive or multiplicative rotational invariant perturbation. We establish, using the Replica method, the asymptotic global law estimate for three general classes of noisy matrices, significantly extending previously obtained results. We give exact results concerning the asymptotic deviations (called overlaps) of the perturbed eigenvectors away from the true ones, and we explain how to use these overlaps to "clean" the noisy eigenvalues of $\textbf{M}$. We provide some numerical checks for the different estimators proposed in this paper and we also make the connection with some well known results of Bayesian statistics.

Suggested Citation

  • Joel Bun & Romain Allez & Jean-Philippe Bouchaud & Marc Potters, 2015. "Rotational invariant estimator for general noisy matrices," Papers 1502.06736, arXiv.org, revised Oct 2016.
    • Handle: RePEc:arx:papers:1502.06736
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    Cited by:

    1. Christian Bongiorno & Damien Challet, 2021. "Covariance matrix filtering with bootstrapped hierarchies," PLOS ONE, Public Library of Science, vol. 16(1), pages 1-13, January.
    2. Gautier Marti & Frank Nielsen & Miko{l}aj Bi'nkowski & Philippe Donnat, 2017. "A review of two decades of correlations, hierarchies, networks and clustering in financial markets," Papers 1703.00485, arXiv.org, revised Nov 2020.
    3. Silvia Bartolucci & Fabio Caccioli & Francesco Caravelli & Pierpaolo Vivo, 2020. "Upstreamness and downstreamness in input-output analysis from local and aggregate information," Papers 2009.06350, arXiv.org, revised Feb 2024.
    4. Sumanjay Dutta & Shashi Jain, 2023. "Precision versus Shrinkage: A Comparative Analysis of Covariance Estimation Methods for Portfolio Allocation," Papers 2305.11298, arXiv.org.
    5. Christian Bongiorno & Damien Challet, 2020. "Nonparametric sign prediction of high-dimensional correlation matrix coefficients," Papers 2001.11214, arXiv.org.
    6. Huiqin Xin & Sihai Dave Zhao, 2023. "A compound decision approach to covariance matrix estimation," Biometrics, The International Biometric Society, vol. 79(2), pages 1201-1212, June.
    7. Gautier Marti & Philippe Very & Philippe Donnat & Frank Nielsen, 2015. "A proposal of a methodological framework with experimental guidelines to investigate clustering stability on financial time series," Papers 1509.05475, arXiv.org.
    8. Christian Bongiorno & Marco Berritta, 2023. "Optimal Covariance Cleaning for Heavy-Tailed Distributions: Insights from Information Theory," Papers 2304.14098, arXiv.org, revised Apr 2023.
    9. Florent Benaych-Georges & Nathanaël Enriquez & Alkéos Michaïl, 2019. "Empirical Spectral Distribution of a Matrix Under Perturbation," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1220-1251, September.
    10. Emmanuelle Jay & Thibault Soler & Eugénie Terreaux & Jean-Philippe Ovarlez & Frédéric Pascal & Philippe De Peretti & Christophe Chorro, 2019. "Improving portfolios global performance using a cleaned and robust covariance matrix estimate," Documents de travail du Centre d'Economie de la Sorbonne 19022, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    11. Bongiorno, Christian & Challet, Damien, 2023. "Non-linear shrinkage of the price return covariance matrix is far from optimal for portfolio optimization," Finance Research Letters, Elsevier, vol. 52(C).
    12. Feng, Wenjun & Zhang, Zhengjun, 2023. "Risk-weighted cryptocurrency indices," Finance Research Letters, Elsevier, vol. 51(C).
    13. Ding, Xiucai & Ji, Hong Chang, 2023. "Spiked multiplicative random matrices and principal components," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 25-60.
    14. Emmanuelle Jay & Thibault Soler & Eugénie Terreaux & Jean-Philippe Ovarlez & Frédéric Pascal & Philippe de Peretti & Christophe Chorro, 2019. "Improving portfolios global performance using a cleaned and robust covariance matrix estimate," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-02354596, HAL.
    15. Yan Zhang & Jiyuan Tao & Zhixiang Yin & Guoqiang Wang, 2022. "Improved Large Covariance Matrix Estimation Based on Efficient Convex Combination and Its Application in Portfolio Optimization," Mathematics, MDPI, vol. 10(22), pages 1-15, November.
    16. Emmanuelle Jay & Thibault Soler & Eugénie Terreaux & Jean-Philippe Ovarlez & Frédéric Pascal & Philippe de Peretti & Christophe Chorro, 2019. "Improving portfolios global performance using a cleaned and robust covariance matrix estimate," Post-Print halshs-02354596, HAL.
    17. Charles-Albert Lehalle & Guillaume Simon, 2021. "Portfolio selection with active strategies: how long only constraints shape convictions," Journal of Asset Management, Palgrave Macmillan, vol. 22(6), pages 443-463, October.

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